Estimation of the directional parameter of the offset exponential and normal distributions in three-dimensional space using the sample mean

TL;DR

This paper analytically derives the directional precision of the sample mean estimator for offset exponential and normal distributions in 3D, revealing convergence behaviors and providing approximation formulas.

Contribution

It introduces explicit formulas for the directional precision of the sample mean for offset exponential and normal distributions in three-dimensional space.

Findings

Sample mean of shifted exponential relates to Bessel and hypergeometric functions.

Distribution of the sample mean converges to normal near the mode.

Provided approximation formulas for directional precision and shift estimation.

Abstract

The directional precision of the sample mean estimator was calculated analytically for the offset exponential and normal distributions in three-dimensional space both for a finite sample and for limiting cases. It was shown that the spherical projection of the sample mean of the shifted exponential distribution has connections with modified Bessel functions and with hypergeometric functions. It was shown explicitly how the distribution of the sample mean of the exponential pdf converges near the mode to the normal distribution. Approximation formulae for the distribution of the sample mean of the shifted exponential distribution and for its directional precision and for the precision of the estimation of the direction of shift of the normal distribution were obtained.